Where are we?

Unless something like the Bohmian interpretation or a spatial collapse theory is right, quantum mechanics gives us good reason to think that the position wavefunction of all our particles is spread across pretty much all of the observable universe. Of course, except in the close vicinity of what we pre-theoretically call “our body”, the wavefunction is incredibly tiny.

What are we to make of that for the “Where am I?” question? One move is to say that we all overlap spatially, occupying most of the observable universe. On a view like this, we better not have position do serious metaphysical or ethical work, such as individuating substances or making moral distinctions based on whether one individual (say, a fetus) is within the space occupied by another.

The other move is to say I am where the wavefunction of my particles is not small. On a view like this, my location is something that comes in degrees depending on what our cut-off for “small” is. We get to save the intuition that we don’t overlap spatially. But the cost of this is that our location is far from a fundamental thing. It is a vague concept, dependent on a cut-off. A more precise thing would be to say things like: “Here I am up to 0.99, and here I am up to 0.50.”

From a determinable-determinate model of location to a privileged spacetime foliation

Here’s a three-level determinable-determinate model of spacetime that seems somewhat attractive to me, particularly in a multiverse context. The levels are:

1. Spatiotemporality

2. Being in a specific spacetime manifold

3. Specific location in a specific spacetime manifold.

Here, levels 2 and 3 are each a determinate of the level above it.

Thus, Alice has the property of being at spatiotemporal location x, which is a determinate of the determinable of being in manifold M, and being in manifold M is a determinate of the determinable of spatiotemporality.

This story yields a simple account of the universemate relation: objects x and y are universemates provided that they have the same Level 2 location. And spatiotemporal structure—say, lightcone and proper distance—is somehow grounded in the internal structure of the Level 2 location determinable. (The “somehow” flags that there be dragons here.)

The theory has some problematic, but very interesting, consequences. First, massive nonlocality, both in space and in time, both backwards and forwards. What spacetime manifold the past dinosaurs of Earth and the present denizens of the Andromeda Galaxy inhabit is partly up to us now. If I raise my right hand, that affects the curvature of spacetime in my vicinity, and hence affects which manifold we all have always been inhabiting.

Second, it is not possible to have a multiverse with two universes that have the same spacetime structure, say, two classical Newtonian ones, or two Minkowskian ones.

To me, the most counterintuitive of the above consequences is the backwards temporal nonlocality: that by raising my hand, I affect the level 2 locational properties, and hence the level 3 ones as well, of the dinosaurs. The dinosaurs would literally have been elsewhere had I not raised my hand!

What’s worse, we get a loop in the partial causal explanation relation. The movement of my hand affects which manifold we all live in. But which manifold we all live in affects the movement of the objects in the manifold—including that of my hand.

The only way I can think of avoiding such backwards causation on something like the above model is to shift to some model that privileges a foliation into spacelike hypersurfaces, and then has something like this structure:

1. Spatiotemporality

2. Being in a specific branching spacetime

3. Being in a specific spacelike hypersurface inside one branch

4. Specific location within the specific spacelike hypersurface.

We also need some way to handle persistence over time. Perhaps we can suppose that the fundamentally located objects are slices or slice-like accidents.

I wonder if one can separate the above line of thought from the admittedly wacky determinate-determinable model and make it into a general metaphysical argument for a privileged foliation.

Does general relativity lead to non-locality all on its own?

A five kilogram object has the determinable mass with the determinate mass of 5 kg. The determinate mass of 5 kg is a property that is one among many determinate properties that together have a mathematical structure isomorphic to a subset of real numbers from 0 to infinity (both inclusive, I expect). Something similar is true for electric charge, except now we can have negative values. Human-visible color, on the other hand, lies in a three-dimensional space.

I think one can have a Platonic version of this theory, on which all the possible determinate properties exist, and an Aristotelian one on which there are no unexemplified properties. There will be important differences, but that is not what I am interested in in this post.

I find it an attractive idea that spatial location works the same way. In a Newtonian setting the idea would be that for a point particle (for simplicity) to occupy a location is just to have a determinate position property, and the determinate position properties have the mathematical structure of a subset of three-dimensional Euclidean space.

But there is an interesting challenge when one tries to extend this to the setting of general relativity. The obvious extension of the story is that determinate instantaneous particle position properties have the mathematical structure of a subset of a four-dimensional pseudo-Riemannian manifold. But which manifold? Here is the problem: The nature of the manifold—i.e., its metric—is affected by the movements of the particles. If I step forward rather than back, the difference in gravitational fields affects which mathematical manifold our spacetime is isomorphic to. If determinate position properties are tied to a particular manifold, it means that the position of any massive object affects which manifold all objects are in and have always been in. In other words, the account seems to yield a story that is massively non-local.

(Indeed, the story may even involve backwards causation. Since the manifold is four-dimensional, by stepping forward rather than backwards I affect which four-dimensional manifold is exemplified, and hence which manifold particles were in. )

This is interesting: it suggests that, on a certain picture of the metaphysics of location, general relativity by itself yields non-locality.

Real Presence and primitive locational relations

According to relationalism, space is constituted by the network of spatial relations, such as metric distance relations (e.g., being seven meters apart). If these relations are primitive, then there is a very easy way for God to ensure the Real Presence of Christ: he can simply make there be additional spatial relations between Christ and other material entities, spatial relations that are exactly like the relations that the bread and wine stood in to other material entities.

It might seem contradictory for Christ to stand in two distance relations: for instance, being one mile from me (in one church) and three miles from me (in another). But I doubt this is a contradiction. New York and London are both 5600 km and 34500 km apart, depending on which direction you go.

According to substantivalism, on the other hand, points or regions are real, and objects are in a location by standing in a relation to a point or region. If relations are primitive, again there should be no problem about God instituting additional such relations to make it be that Christ is present where the bread and wine were.

In other words, if location is constituted by a primitive relation—whether to other objects or to space—there is apt to be no difficulty in accounting for the Real Presence. The reason is that we expect, barring strong reason to the contrary, primitive relations to be arbitrarily recombinable.

If location, however, is constituted by a non-primitive relation, there might be more difficulties. For instance, as a toy theory, consider the variant of relationalism on which spatial relations are constituted by gravitational force relations (two objects have distance r if and only if they have masses m₁ and m₂ and there is a gravitational force Gm₁m₂/r² between them). In that case, for God to make Christ present in Waco would require God to make Christ stand in gravitational force relations of the sort that I stand in by virtue of being in Waco. For instance, the earth’s gravitational force on Christ would have to point from Waco to the center of the earth—but since the Eucharist is also in Rome, it would have to point from Rome to the center of the earth as well. And that might be thought impossible. But perhaps there could be two terrestrial gravitational forces on Christ: one along the Waco-geocenter vector and the other along the Rome-geocenter vector. This would require some sort of a realism about component forces, but that’s probably necessary for the gravitational toy theory. And then God would have to miraculously ensure that despite the forces, Christ is not affected in the way he would normally be by these forces. All this may be possible, but it’s less clear than if we have primitive relations.

Inductive evidence of the existence of non-spatial things

Think about other plausibly fundamental qualities beyond location and extension: thought, charge, mass, etc. For each one of these, there are things that have it and things that don’t have it. So we have some inductive reason to think that there are things that have location and things that don’t, things that have extension and things that don’t. Admittedly, the evidence is probably pretty weak.

Location, causation and transsubstantiation

Here’s a fun thought experiment. By a miracle (say) I am sitting in my armchair in Waco but my causal interaction with my environment at the boundaries of my body would be as if I were in Paris. There is a region of space in Paris shaped like my body. When a photon hits the boundary of that region, it causally interacts with me as if I were in Paris: I have the causal power to act at a distance to reflect Parisian photons as if I were in that region in Paris. Alternately, that photon might be absorbed by me: I have the causal power to absorb Parisian photons. As a result, it looks to Parisians like I am in Paris, and as I look around, it looks to me like Paris is all around me. The same is true for other interactions. When my vocal cords vibrate, instead of causing pressure changes in Texan air, they cause pressure changes in chilly France. As I walk, the region of space shaped like my body in Paris that is the locus of my interaction with Parisians moves in the usual way that bodies move.

Furthermore, my body does not interact with the environment in Waco at all. Wacoan photons aimed at my body go right through it and so I am invisible. In fact, not just photons do that: you could walk right through my body in Waco without noticing. My body is unaffected by Texan gravity. It is simply suspended over my sofa. As I wave my hand, my hand does in fact wave in Texas, but does not cause any movement of the air in Texas—but in Paris, the region of space in which I interact with the Parisians changes through the wave, and the air moves as a result. When I eat, it is by means of Parisian food particles that come to be incorporated into my Wacoan body.

To me, to Wacoans and to Parisians it looks in all respects like I am in Paris. But I am in Waco.

Or am I? There is a view on which the causal facts that I’ve described imply that I am in Paris, namely the view that spatial relationships reduce to causal relationships. It is an attractive view to those like me who like reductions.

But this attractive view threatens to be heretical. Christ’s body is here on earth in the Eucharist, as well as in heaven in the more normal way for a body to be. But while the body is surely visible in heaven and interacts with Mary and any other embodied persons in heaven, it does not interact physically with anything on earth. Granted, there is spiritual interaction: Christ’s presence in the Eucharist is a means of grace to recipients. But that probably isn’t the sort of interaction that would ground spatial location.

There is, however, a way to modify the causal reduction of location that handles the case of the Eucharist. Actual causal interactions do not seem to be enough to ground location. The reduction very likely needs needs dispositional causal interactions that typically ground causal counterfactuals like:

1. If Parisians were to shine a flashlight into that dark alley, they’d see me.

However, dispositions can be masked. For instance, sugar is still soluble even if God has promised to miraculously keep it from dissolving when it is placed in water. In such a case, the sugar still has the disposition to dissolve in water, but fails to ground the counterfactual:

2. The lump would be dissolved were it placed in water.

We might, thus, suppose that when the Mass is being celebrated in Waco, Christ comes to have the dispositional causal properties that would ordinarily be contitutive of his being present in Waco, such as the disposition to reflect Texan photons, and so on. But by miracle, all these dispositions are masked and do not result in actual causal interaction. The unmasked dispositions are those corresponding to spiritual interaction.

Here’s an interesting lesson. The kind of causal-reductive view of location that I’ve just considered seems to be one of the least transsubstantiation-congenial views of location. But, nonetheless, the transsubstantiation can still be made sense of on that view when the view is refined. This gives us evidence that transsubstantiation makes sense.

And we can now go back to the story of my being in Waco while interacting in Paris. The story was underspecified. I didn’t say whether I have the dispositions that go with being in Waco. If I do, these dispositions are being miraculously masked. But they may be enough to make me count as being in Waco. So on the story as I’ve told it, I might actually be both in Waco and in Paris.

Final question: Can external temporal location be similarly causally grounded? (Cf. this interesting paper.)

Are there unicorns here?

Multiverse theories like David Lewis’s or Donald Turner’s populate reality with a multitude of universes containing strange things like unicorns and witches riding broomsticks. One might think that positing unicorns and witches makes a theory untenable, but the theorists try to do justice to common sense by saying that the unicorns and witches aren’t here. Each universe occupies its own spacetime, and the different spacetimes have no locations in common.

But why take the different universes to have no locations in common? Surely, just as a unicorn can have the same charge or color as I, it can have the same location as I. From the fact that a unicorn can have the same charge or color as I, we infer in a Lewisian setting that some unicorn does have the same charge or color as I (and likewise in Turner’s, with some plausible auxiliary assumptions about values). Well, by the same token, from the fact that a unicorn can have the same location as I, we should be able to infer that some unicorn does have the same location as I.

Not so, says Lewis. Counterpart theory holds for locations, but not for charges and colors. What makes it true that a unicorn can have the same charge as I now have is that some unicorn does have the same charge as I. But what makes it true that a unicorn can have the same location as I now have is that some unicorn has a counterpart of my location in a different spacetime.

But what justifies this asymmetry between the properties of charge and location? The asymmetry seems to require clauses in Lewis’s modal semantics that work differently for different properties. It seems there are properties—say, being green—whose possible possession is grounded in something’s having the property, and there are properties—say, being at this location—whose possible possession is grounded in something’s having a a counterpart of the property.

Specifying in a non-ad hoc way which properties are which rather complicates the system. Moreover, it leads to this oddity. Lewis thinks abstract objects exist in all worlds. So, he has to say that being at this location exists in all worlds. And yet the counterpart of being at this location in another world is a different property, even though this exact property does exist at that world.

There is a solution for Lewis. Lewis is committed to counterpart theory holding for objects. It is reasonable for him thus to take counterpart theory also to hold for properties defined de re in terms of particular non-abstract objects. Thus, what makes it true that a unicorn could have had the property of being a mount of Socrates is not that some unicorn in some universe has this property—for no unicorn in our universe has that property, and Socrates according to Lewis only exists in our world—but that some unicorn has a counterpart to this property, which counterpart property is the property of being a mount of S where S is a counterpart of Socrates.

If Lewis can maintain that location properties are defined de re by relation to non-abstract objects, then he has a way out of the objection. Two kinds of theories allow a Lewisian to do this. First, the Lewisian can be a substantivalist who thinks that points or regions of space are non-abstract. Then being here will consist in being locationally related to some point or region L, and Lewis can take counterpart theory to apply to points or regions L. Second, the Lewisian can be a relationalist and say that location is defined by relations to other physical objects, in such a way that if all the objects were numerically different from what they are, nothing could be in the same place it is, and counterpart theory is applied to physical objects by Lewis.

What Lewis cannot do, however, is take a view of location that either takes location to be a relation to abstract objects—say, sets of points in a mathematical manifold—or that takes location to simply be a non-relational determinable like charge or rest mass.

In particular, multiverse theorists like Lewis and Turner are committed to treating location as different from other properties. Anecdotally, most philosophers do treat location like that. But for those of us who are attracted to the idea that location is just another determinable, this is a real cost.

Partial location, quantum mechanics and Bohm

The following seems to be intuitively plausible:

1. If an object is wholly located in a region R but is not wholly located in a subregion S, then it is partially located in R−S.
2. If an object is partially located in a region R, then it has a part that is wholly located there.

The following also seems very plausible:

3. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is 1, then the particle is wholly located in the closure of R.
4. If the integral of the modulus squared of the normalized wavefunction for a particle over a region R is strictly less than one, then the particle is not wholly located in the interior of R.

But now we have a problem. Consider a fundamental point particle, Patty, and suppose that Patty's wavefunction is continuous and the integral of the modulus squared of the wavefunction over the closed unit cube is 1 while over the bottom half of the cube it is 1/2. Then by (3), Patty is wholly contained in the cube, and by (4), Patty is not wholly contained in the interior bottom half of the cube. By (1), Patty is partially located in the closed upper half cube. By (2), Patty has a part wholly located there. But Patty, being a fundamental particle, has only one part: Patty itself. So, Patty is wholly located in the closed upper half cube. But the integral of the modulus squared of the wavefunction over the closed upper half cube is 1−1/2=1/2, and so (4) is violated.

Given that scenarios like the Patty one are physically possible, we need to reject one of (1)-(4). I think (3) is integral to quantum mechanics, and (1) seems central to the concept of partial location. That leaves a choice between (2) and (4).

If we insist on (2) but drop (4), then we can actually generalize the argument to conclude that there is a point at which Patty is wholly located. Either there is exactly one such point--and that's the Bohmian interpretation--or else Patty is wholly multilocated, and probably the best reading of that scenario is that Patty is wholly multilocated at least throughout the interior of any region where the modulus squared of the normalized wavefunction has integral one.

So, all in all, we have three options:

• Bohm
• massive multilocation
• partial location without whole location of parts (denial of (2)).

This means that either we can argue from the denial of Bohm to a controversial metaphysical thesis: massive multilocation or partial location without whole location of parts, or we can argue from fairly plausible metaphysical theses, namely the denial of massive multilocation and the insistence that partial location is whole location of parts, to Bohm. It's interesting that this argument for Bohmian mechanics has nothing to do with the issues about determinism that have dominated the discussion of Bohm. (Indeed, this argument for Bohmian mechanics is compatible with deviant Bohmian accounts on which the dynamics is indeterministic. I am fond of those.)

I myself have independent motivations for embracing the denial of (2): I believe in extended simples.

Degrees of location?

If collapse versions of quantum mechanics are right, then objects typically don't have location simpliciter. Instead, they have a wavefunction the square of whose modulus describes the probability that the object will collapse to a given location. Perhaps right after a collapse, the objects have a single definite location, but the single definite location doesn't last beyond that moment.

Suppose we take all this seriously as metaphysics. I think there are several options. The first is that we should take the wavefunction, or the square of its modulus, as providing the whole story about an object's location. In that case, it is rarely if ever correct to say that an object has a particular location. Instead, one should say that objects have their locations to various degrees. (This degreedness of location is different from the way in which we can say that a person who has one leg in a room is to a lesser degree in the room than someone who has an arm and a leg in it.) Location, at least as exhibited in the actual world, is a degreed property.

The second option is that an object is wholly in a location when and only when the wavefunction assigns unit probability to its being there. This has the odd consequence that if a point particle is wholly in a region R, then it is also in every punctured region of the form R−{x}, since the integral of the modulus squared of the wavefunction over R and over R−{x} will be the same. But this means that the particle is wholly absent from every point of the region R, even while wholly present in the region R. That seems problematic.

A third option is that being wholly located is metaphysically primitive, and there is a law of nature that makes it be the case that when the integral of the modulus squared of the (normalized) wavefunction of a particle over a region R is 1, and the region R is "nice" (e.g., equals the interior of its closure), the particle is wholly in R.

I like the first option most...

Two conceptions of matter

The philosophical tradition contains two conceptions of matter. One kind, associated with Descartes, connects matter with space: matter is what is responsible for spatial properties like extension or location. The other, associated with Aristotle, connects matter with passivity: matter is what makes an entity have a propensity to be the patient of causal influences. The spatial conception of matter has been the more popular one in recent times. But here is a reason not to go for the spatial conception of matter. The concept of materiality seems fairly close to the fundamental level. But it may well turn out--string theory is said to push in that direction--that at the fundamental level there is no such thing as space or time or spacetime. If that is a serious epistemic possibility, it would be good to do more work on the Aristotelian option.

Spacetime: Beyond substantivalism and relationalism

According to substantivalism, spacetime or its points or regions is a substance, and location is a relation between material things and spacetime or its points or regions. According to relationalism, location is constituted by relations between material things. Often, the two views are treated as an exhaustive division of the territory.

But they're not. Lately, I've found myself attracted to a tertium quid which I know is not original (it's a story other people, too, have come to by thinking about the analogy between location and physical qualities like charge or mass). On a simplified version of this view, being located is a determinable unary property. Locations are simply determinates of being located. This picture is natural for other physical qualities like charge. Having charge of 7 coulombs is not a matter of being related to some other substances--whether other charged substances or some kind of substantial "chargespace" or its points or regions. It's just a determinate of the determinable having charge.

This determinate-property view is more like the absolutism of substantivalism, but differs from substantivalism by not positing any "spacetime substance", or by making the locations into substances. Locations are determinates of a property, and hence are properties rather than substances. If nominalism is tenable for things like charge or mass, the theory won't even require realism about locations.

What is a material object?

I've found the notion of a material object very puzzling. Here is something that would render it less puzzling to me:

• x is a material object if and only if x has limited location.

There would then be three ways for an object y to be immaterial:

1. There are locations and y has no location.
2. There are no locations.
3. There are locations and y is unlimited in location.

It would now be plausible that a perfect being would be necessarily immaterial. A perfect being doesn't need anything other than itself, so it could exist in worlds where there are no locations, in which worlds it would have type 2 immateriality. And in worlds where there are locations, a perfect being would be unlimited in location, and would have type 3 immateriality. Thus, in all worlds, a perfect being would have immateriality. But in no world would a perfect being have type 1 immateriality.

One might worry that there could be an animal that is as big as space itself, and then it would count as an immaterial object. But even though the animal would be everywhere, it wouldn't be everywhere in every part and respect. Its digestive system would be here but not there, and so on.

Alternately, one might stick to our definition of materiality as limited location, but modalize. Maybe "limited location" is a modal concept, so that a being that could be limited in location is thereby limited in location.

What is location?

I will first argue that location is a multiply-realizable—i.e., functional—determinable. Then I will offer a sketch of what defines it.

A multiply-realizable determinable is one such that attributions of its determinates are grounded in different ways in different situations. For instance, running a computer program is multiply realizable: that something is running some algorithm A could be at least partly made true by electrical facts about doped silicon, or by mechanical facts about gears, or by electrochemical facts about neurons. Moreover, computer programs can run in worlds with very different laws from ours.

As a result, a multiply-realizable determinable is not fundamental. But location seems fundamental, so what I am arguing for seems to be a non-starter. Bear with me.

Consider a quantum system with a single particle z. What does it mean to say that z is located in region A at time t?[note 1] It seems that the quantum answer is: The wavefunction (in position space) ψ(x,t) is zero for almost all x outside A. And more generally, quantum mechanics gives us a notion of partial location: x is in A to degree p provided that p=∫_A|ψ(x,t)|²dx, assuming ψ is normalized. On these answers, being located in A is not fundamental: it is grounded in facts about the wavefunction.

But it is also plausible that objects that do not have wavefunction can have location. For instance, there may be a world governed by classical Newtonian mechanics, and objects in that world have locations but no wavefunctions. (And even in a world with the same laws as ours, it is possible that some non-quantum entity, like an angel, might have a location, alongside the quantum entities.) Thus, location is multiply-realizable.

Very well. But what is the functional characterization of location? What makes a determinable be a location determinable? A quantum particle is located in A provided that ψ vanishes outside A. But a quantum particle also has a momentum-space wavefunction, and we do not want to say that it is located in A provided that the momentum-space wavefunction vanishes outside A? Why is the "position-space" wavefunction the right one for defining location? Why in a classical world is it the "position" vector that defines location, rather than, say, the momentum vector or an axis of spin or even the electric charge (a one-dimensional position)?

I want to suggest a simple answer. Two objects can have very similar electric charges, very similar spins or very similar momenta, and yet hardly be capable of interacting because they are too far apart. In our world, distance affects the ability of objects to interact with one another. Suppose we say that this is the fundamental function of distance. Then we can say that a determinable L is a location-determinable to the extent that L is natural and the capability of objects to interact with one another tends to be correlated with the closeness of values of L. This requires that L have values where one can talk about closeness, e.g., values lying in a metric space. In a quantum world without too much entanglement and with forces like those in our world, the wavefunction story gives such a determinable. In a classical world, the position gives such a determinable.

(One could also have an obvious relationalist variant, where we try to define the notion of being spatially related instead. The same points should go through.)

Notice that on this story, it may be vague whether in a world some determinable is location. That seems right.

I think this story fits well with common-sense thought about distance and location, and helps explain why we maintained these concepts across radical changes in physical theory.

This story also reminds me a little of what Aquinas says about the location of angels: An angel is at location x if and only if the angel is causally interacting with something at x. But there is a difference. While Aquinas defines location of an angel in terms of actual interaction, I define location at two removes from interaction: I only talk of the capability for interaction and I do not define location in terms of that, but in terms of a fairly natural determinable closeness in respect of which tends to correlate with capability for interaction.

We have two non-philosophical tests for a theory of location. One is whether it coheres with science and the other is whether it coheres with theology, and especially with transsubstantiation. It is no surprise that this theory coheres with science, since it was designed to. What about transsubstantiation (John Heil referred to it as a supercollider for metaphysical theories in medieval times)?

I think it coheres with transsubstantiation quite well. The Catholic tradition tends not to talk about physical but sacramental presence. This is a real presence of course. There are multiple ways of being located for Aquinas: physically, by power (as angels are where they act and as God is everywhere), as well as this sacramental presence that he has an elaborate metaphysical theory of (see my paper here). Our above theory allows for a multiplicity of location determinables, all taking values in a common space, and so there might be some determinable that would give an account of sacramental presence.

But one could also defend the physical presence of Christ's body in the Eucharist coherently with my view. Christ's body could have multiple wavefunctions, each defining a different location. Or maybe space could curve in on itself as I suggest (but do not endorse as my preferred view) in the above-cited paper so that the location of the consecrated host and the location of heaven are literally the same.

So while I didn't design the view initially with the transsubstantiationon in mind, I think it passes the transsubstantiation test.

Internal space

As David Lewis taught us, time travel calls for a notion of internal time. If I am about to travel to the time of the dinosaurs, then maybe in an hour I will meet a dinosaur. But that's an internal time hour. If I am going to spend the rest of my life in the Mesozoic, then—assuming nothing kills me—I will grow old before I am born, but this "before" is tied to external time, since of course in internal time, I grow old after being born.

Perhaps ordinary travel calls for a notion of internal space. Let's say today I am in room 304 of the hospital, and yesterday I was in room 200. The doctor comes and asks: "Does it still hurt in the same place as it did yesterday?" I tell her: "No, because yesterday it hurt in room 200, and today it hurts in room 304." But that's external place, and the doctor was asking about internal place.

Internal place is moved relative to external place while the body as a whole is locomoting. But it can also be moved when only parts of the body are moving. If my hands are hurting, and I clasp my hands to each other, I thereby make the internal places where it hurts be very close externally, but they are still as distant internally as they would be were I to hold my arms wide. If, on the other hand, my two hands grew together into a new super-hand, the two places would come to be close together.

I wonder: If I grow, does my head come to be internally further from my feet? I think so: There are more cells in between, for instance.

Rob Koons has suggested to me that the notion of internal place can help with Brentano's notion of "coincident boundaries": Suppose we have two perfect cubes, with the red one on top of the green one. Then it seems that the red cube's bottom boundary is in the same place as the green cube's top boundary. (Sextus Empiricus used basically this as an argument against rigid objects.) Question: But how can there two boundaries in the same place? Answer: There are two internal places in one external place here.

"Wholly present"

Here's something I've been thinking about. I want to start with the technical notion of being located at a region. This notion allows for partial location. If I have one leg in Arkansas and one in Texas, then I count as located in Arkansas and located in Texas. If regions have points, then I am located at a region if and only if I located at some point in the region (maybe that's a more primitive notion).

I'd like to move from the notion of being located at a region to the notion of being wholly present in a region. I am now wholly located in Texas, but if I had a leg in Arkansas and a leg in Texas, I would be wholly located in neither.

I could take the notion of being wholly present as primitive. But I don't want to do because it's a three-dimensional notion, while I think I am a four-dimensional entity, so to me it's a derivative notion.

One obvious thing to say is:

1. x is wholly present in A iff x is located at A and not located outside A.

This rules out multilocation—being wholly present at two distinct regions—by fiat. In so doing, it rules out both my view of the doctrine of transsubstantiation (since on my view, it is literally true that Christ is wholly spatially present in different places[note 1]) and the possibility of backwards time travel (since if you can travel back in time, you could be wholly present in two places, and shake hands with yourself).

A plausible move is to introduce parts or, more generally, features (the blueness of my eyes is a feature but not a part) and their locations (I stipulate that every part, proper or not, is a feature). Maybe, then:

2. x is wholly present in A iff every feature of x is located at A.

While this works for transsubstantiation, it doesn't work for time travel. For suppose that tomorrow I lose a leg, and I travel back to today, so that I am in another room in addition to this one. Then it is false that every part of me is located in that other room, since the lost leg isn't there.

There is another interesting problem with (2) and time travel. Suppose that in ten seconds I travel back to the present, so that I am wholly present in two disconnected rooms, and suppose that in the ten seconds I have neither lost nor gained any features. Let A_L be the left half of the space occupied by me in one of the rooms and B_R be the right half (including the cut line—don't put the cut line in A_L) of the space occupied by me in the other room. Let C be the disconnected region that is the union of A_L and B_R. Then by (2) I am wholly present at C as every one of my features is in either A_L or in B_R or in both. But surely I am not wholly located in the messy region C.

At this point things get difficult. My best solution today is moderately complex (but not as complex as my best solution yesterday). It requires the introduction of two sets of times for a persisting substance. There are internal times, which correspond to the internal development of the substance, and there are external times, which correspond to what goes on in the external world. Normally, the two are nicely correlated. But time travel discombobulates things. If in one minute I travel 24 hours into the future, then in one internal minute I will be 24 external hours forward. And if in a minute I travel 24 hours into the past, then in one internal minute it will be externally yesterday.

Now, take the case where I am right now in room A in the normal way, but in room B due to having time-traveled back to that room after losing a leg. Let T be the present external time. There are two internal times associated with t. At internal time t₁, I am in room A, and at internal time t₂, I am in room B. Moreover, at t₁, I have two legs, though at t₂ I only have one. I guess at the external time T, I have two legs. My being wholly present in B does not require that I have both of my legs in B. It only requires that I have in B all the legs that I have at the internal time t₂.

This yields the following pair of definitions:

3. x is wholly present in A at its internal time t iff every feature that x has at t is located at A at t.
4. x is wholly present in A at external time T iff there is an internal time t of x such that (a) x's internal time t is externally at T and (b) x is wholly present in A at t.

This gives the right answers with respect to (a) transsubstantiation, (b) time-travel and loss/gain of parts, and (c) time-travel and the union of the A_L and B_R regions.

The account does, however, have the consequence that if x is an extended simple with all features spread over all of x (so, x is the same color all over, etc.), then it counts as wholly present at every point at which it is located. This consequence is perhaps not so plausible, but I can live with it.

Adverbial ontology and dispensing with parts

Once one has an adverbial ontology, like the one I used to help with the Incarnation, one no longer needs the parts of a substance in one's ontology. "I have two hands." That's made true by my being two-handed. "My right hand has five fingers." That's made true by my having a right hand five-fingeredly. More explicitly, there is a mode m in virtue of which I have a right hand. (According to my Incarnation post, I have m indirectly, as m is a mode of my humanity.) Then we have two moves we can make. We could say that there are five modes of m, which each of which is a different way of the hand's being fingered. If we go that route, then we are forced into identity of indiscernibles for fingers, and by extension for any other parts. I welcome that consequence myself, since I'm anyway pulled to identity of indiscernibles by my theory of transworld identity. But alternately we could simply posit a mode of being five-fingered, perhaps a mode relational to the number five.

To my mind there are three main uses of parts:

1. Some properties of wholes are grounded in properties of parts. "I have the property of having heart-beat in virtue of having as a part a heart that in turn has the property of beating."
2. Parts have location and help explain partial location. "I am partly in this room and partly in that, because one of my legs is here and the other there."
3. Some parts are widely thought to be able to move between substances. "Several hours after you ate the apple, a carbon atom that used to be a part of an apple tree has become a part of you."

The adverbial mode ontology does the first two tasks well.

1: There is a mode in virtue of having which I have heart-beat. But I have that mode indirectly: that mode modifies my being hearted, which in turn modifies my humanity. So properties are divided up. But an advantage of the mode way is that we get to uniformly divide properties not just by parts, but by functional subsystems. Some functional subsystems correspond to parts, but likely not all. In a computer running several processes at once on the same processor core, the processes may correspond to different functional systems—say, one doing Fourier transforms of microphone data and another watching for user input events—but the processes may be implemented by overlapping sets of physical parts (and a computer has no others), and we could easily imagine that there is no distinct set of parts corresponding to each process. It seems likely that something like that happens in the brain, and even if it does not, the possibility should be accounted for in our ontology. The adverbial mode ontology apportions properties had in virtue of a functional subsystem in the same way that it apportions those properties had in virtue of a physical part, and that strikes me as exactly right.

2: This is really just a special case of 1. "My right leg is located in this room" is true in virtue of my being right-leg-possessed this-roomly.

But unless we posit that modes can move between substances—I've heard Rob Koons speculate in that direction and Aquinas's account of transsubstantiation famously allows modes to survive the destruction of their underlying substance—it's harder to handle 3. On general Aristotelian grounds, I think one can just bite the bullet. The identity of a part, if there are parts, is going to be dependent on the whole. There is no carbon atom that was a part of the apple tree and is now a part of you. There are (in the eternalist sense of "are"), at best, two carbon atoms, one that was identity dependent on the tree and the other that is identity dependent on you, and the first caused the second. This is counterintuitive.

So what our adverbial ontologists should say about 3 is that the apple tree has some mode m₁ that makes it count as having had a certain carbon atom once, and you have some mode m₂ that makes you count as having a certain carbon atom. There is a continuity of location (see 2) between the one mode (perhaps with some other intervening modes, depending on the ontological status of the apple as such) and the other. Moreover, m₁ is a cause of m₂. Or, if we prefer (and I think we should), the apple tree as modified by m₁ caused you to have m₂. I.e., there is a mode c₁ of causation had by m₁, which is a causation of m₂, or of me as having m₂. But the numerical identity of the particles, that we need to give up on. However, since giving up on parts dissolves the problem of material constitution, and since every other solution to the problem of material constitution has something else counterintuitive about it, we are in this regard no worse off here than any view on which there are parts.

A challenge for the view is to distinguish between those modes that correspond to parts and those modes that don't. But one might just reject the distinction. Or one might go like this. It might be that all and only the modes that have a location mode are parts. But don't non-part subsystems have a location mode? Maybe not. Rather, they may be modes--or joint modes (maybe a mode can be a mode of more than one mode--or maybe even more than one substance--and maybe that is how relations are to be handled)--of one or more parts, and the parts are what have a location mode. The non-part subsystems, then, have a location in a derivative sense.

More about functionalism about location

Functionalism about location holds that any sufficiently natural relation, say between objects and points in a topological space, that has the right formal properties (and, maybe, interacts the right way with causation) is a location relation.

Here is an argument against functionalism. Functionalism is false for other fundamental physical determinables: it is false for mass, charge, charm, etc. There is a possible world where some force other than electromagnetic is based on a determinable other than charge, but where the force and determinable follow structurally the same laws. By induction, functionalism is probably false for location.

Some will reject this argument precisely because they accept something like functionalism for the other physical determinables, and hence deny the thought experiment about the non-electromagnetic force--they will say that if the laws are structurally the same, the properties are literally the same.

I think there is a way to counter the above argument by pointing out a disanalogy between location and other fundamental physical determinables (this disanalogy goes against the spirit of this post, alas). Let's say we live in an Einsteinian world. A Newtonian world still might have been actual. But, plausibly, the Newtonian world's "mass" is a different determinable from our world's mass. Here's why. In our world, mass is the very same determinable as energy (one could deny this by making it a nomic coextensiveness, but I like the way of identity here). In the Newtonian world "mass" is a different determinable from "energy". Therefore either (a) Newtonian "mass" is a different determinable from mass, or (b) Newtonian "energy" is a different determinable from energy, or (c) both (a) and (b). Of these, the symmetry of (c) is pleasing. More generally, it is very plausible that fundamental physical determinables like mass-energy, charge, charm or wavefunction are all law bound: you change the relevant laws (namely, those that make reference to these determinables) significantly, and you don't have instances of these determinables.

But location does not appear to be law bound. "Location" in a Newtonian spacetime and a relativistic spacetime are used univocally. You can have a set of really weird laws, with a really weird 2.478-dimensional space (for fractional dimensions, see, e.g., here), and yet still have location. Maybe there are some formal constraints on the laws needed for locations to be instantiated, but intuitively these are lax.

Plausibly, natural (in the David Lewis sense of not being gerrymandered) physical determinables that are not law bound are functional. If location is a natural physical determinable, which is very plausible on an absolutist view of spacetime, then it is, plausibly, functional. I think an analogous argument can be run on relationism, except that the fundamentality constraint is a bit less plausible there.

One might question the claim that natural physical determinables that are not law bound are functional. After all, if the claim is plausible with the "physical", isn't it equally plausible without "physical"? But the dualist denies the claim that natural determinables that are not law bound are functional. For instance, awareness seems to be a natural determinable (whose determinates are of a form like being aware of/that ..., and nothing else), but the dualist is apt to deny that it's functional.

In any case, one interesting result transpires from the above. It is an important question whether location is law bound. If we could resolve that, we would be some ways towards a good account of spacetime (if it is law bound, proposals like this one might have some hope, if based on a better physics). The account I give above of law boundedness is rather provisory, and a better account is also needed.

A reduction of spatial relations to an outdated physics

Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time t between distinct particles a and b as (m_am_b/F_ab)^1/2, where F_ab is the magnitude of the gravitational force of a on b at t, and m_a and m_b are the masses at t of a and b respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between a and a to be zero. For every t, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let S be a three-dimensional Euclidean space. Let P be the set of all particles. Let T be the real line. For each object a in P, let T_a be the set of times at which a exists, and let m_a(t) be the mass of a at t. For any pair of objects a and b and time t in both T_a and T_b, let F_ab(t) be the magnitude of the gravitational force of a on b at t. Let Q be the set of all pairs (a,t) such that t is a member of T_a. Say that a function f from Q to S is an admissible position function provided that:

1. If t is a member of both T_a and T_b, then F_ab(t)=m_a(t)m_b(t)/|f(b,t)−f(a,t)|².
2. f''(a,t) is equal to the sum over all particles b distinct from a of (f(b,t)−f(a,t))F_ba(t)/(m_a(t)|f(b,t)−f(a,t)|).

The laws can then be taken to say that the world is such that there is an admissible position function. We can then relativize talk of location to an admissible position function, which plays the role of a reference frame: the location of a relative to f at t is just f(a,t).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).

Occupation is just a relation

That's my new slogan.  It's aimed at philosophical views on which there is something ontologically special about occupying a location in space or spacetime.  But surely to occupy a location in spacetime is just to stand in some sort of a relation (to a location or to other objects).

Consider for instance claims like the following that many mereologists like:

• If R is any region all the points of which are occupied by x, and R* is any sub-region, then there is a part of x that occupies all and only the points in R*.

(We can also talk without invoking points, but points will be convenient.)  Consider now some parallels for other relations:

• If R is a set of propositions all the members of which are believed by x (think of belief as epistemic occupation!), and R* is any subset, then there is a part of x that believes all and only the propositions in R*.
• If R is a set of people all members of which x is a friend to, and R* is any subset, then there is a part of x such that that part is a friend to all and only the people in R*.

These are absurd, though we may non-literally talk that way.  "The part of Josh that is friends with Trent likes epistemology."

Or consider some claim that nothing can be in two places at once.  Make the claim precise, for instance in the following way:

• It is not possible that there is an object x and disjoint regions R and R* such that every part of x occupies some point (perhaps different points for different parts) in R and every part of x occupies some point (ditto) in R*.

But why think that the occupation relation satisfies this kind of an axiom?  

Here's a broad sweeping thought: Otherwise Humean philosophers who believe in all sorts of very general rearrangement principles for fundamental relations do not extend the same courtesy to the occupation relation.  

Ironically, while I am not happy with general recombination principles (that say that any recombination of possible objects makes for a possible scenario), I am happy to allow for wild and crazy rearrangements of the occupation relation--objects being in more than one place at a time, objects occupying spatiotemporally disconnected regions, etc.  If I thought there were such things as parts, I might even be open to such options as composite objects occupying locations that none of their proper parts occupy, parts occupying locations that the whole does not occupy, etc.